Infinity, parts 1 & 2

Infinity, part 1

Last week in "Approaching Infinity," we talked about words. "Finite" comes from the root "fin," which means "end" in French. "Finite" means having an end. "INfinite" means not having an end, or being endless. We also brainstormed words meaning "really big": big, large, gigantic, enormous, etc.. Some words that go along with "infinity" are "endless," "forever," and "eternity."

We identified some of the finite things in the room: a paperclip, our bodies, a pencil. We made the infinity symbol out of human bodies:

I told a story about a mountain made of sand and a bird that carries away one grain of sand every thousand years. When the mountain is gone, eternity has barely even started. Here's a fun version of the story as told by Neil Gaiman.

We started set theory. A set is a container that holds things. Some examples:

{1,2,3}

{nose, Peter, tornado}

We learned how to draw the funny "curly brackets" that surround the set. We also looked at the infinite set of all the counting numbers:

{1,2,3,...}

The "..." part means we keep going forever. We listed some other numbers in this set, like 20, and a googleplex, and one hundred, to make sure all the kids understood the "..." part. This set is named "omega," for which I will write w:

w={1,2,3,...}.

I ended class with the following questions to think about for homework:

• Which is bigger, w or {1,2,3,...,w}?
• For infinite sets, what does "bigger" even mean?

Infinity, part 2

A set is a container. Today we made sets - that is, containers that could hold numbers or other things. The sets included a boat, some origami, and several drawings. Sets can contain sets!

Along the way, it was discovered that if you hold paper and an origami crane over the heating vent, the air makes the paper become "magnetic" and stick to the bird:

We looked at the sets

{1,2,3,4,5,...}

{2,4,6,8,10,...}

and agreed that these sets are the same size, since you can make the numbers "hold hands" (Stanley's expression): 1 and 2 hold hands, 2 and 4 hold hands, and so on, until all the numbers get matched up. Similarly, the sets

{1,2,3,4,5,...,w}

{1,2,3,4,5,...}

are the same size: have w hold hands with 1, then 1 holds hands with 2, and so on, until everybody gets matched up!

{w,1,2,3,4,5,...}

{1,2,3,4,5,...}

The blue squiggly line on the board is inspired by Escher's endless stair.

We also tried to make the endless stair out of bodies. There was mixed success.